[R] distance coefficient for amatrix with ngative valus
R. Michael Weylandt
michael.weylandt at gmail.com
Tue Oct 4 21:59:20 CEST 2011
You are, of course, entirely correct and, once again, I tip my hat to
the erudition of those who comment on this list. My initial
formulation, for a distance on a normed space inherited from the norm,
stands trivially, but as you rightly point out, I'm excluding many
interesting and possibly useful norms.
Follies of youth and all that....
Michael
On Tue, Oct 4, 2011 at 2:06 AM, Rolf Turner <rolf.turner at xtra.co.nz> wrote:
> On 04/10/11 17:05, R. Michael Weylandt wrote:
>
> <SNIP>
>>
>> More importantly, as I said in my initial response, any distance
>> metric worth its salt is translation invariant.
>
> <SNIP>
>
> Point of order, Mr. Chairman. (This is really *toadally* off topic;
> my apologies, but I couldn't resist --- I trained as a pure mathematician).
>
> A *metric* need not in general be translation invariant. Indeed a metric
> need not be defined on a space in which translation makes any sense.
>
> A metric defined in terms of a *norm* (on a normed vector space) by
> rho(x,y) = ||x - y|| is of course by definition translation invariant, and
> that's
> what most of us think in terms of.
>
> But there are perfectly ``reasonable'' metrics, defined on vector spaces,
> which are not translation invariant. Whether these are ``worth their salt''
> is I suppose a matter of taste. (You should pardon the expression. :-) )
>
> A simple e.g. of a non-translation-invariant metric is
>
> rho(x,y) = |x - y|/(1 + |x| + |y|)
>
> (defined on the real line). It is easily checked that rho(.,.) satisfies
> the
> four conditions that a metric must satisfy. (Exercise for the interested
> reader.)
>
> Note that rho(1,2) = 1/4 but rho(2,3) = 1/6, ergo not translation
> invariant.
>
> cheers,
>
> Rolf Turner
>
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